I wonder if any StataCorp people on the list could respond to this
query [why qreg does not allow factor variables] with a definitive
answer?

We at StataCOrp are working on making -qreg- with factor variables.
Maarten Buis <maartenlbuis@gmail.com> speculated

-qreg- seems to have its own build-in maximization algorithm, which
does not surprise me as it is a maximization problem with its own
peculiar set of difficulties. So it is a lot more work to implement
it.